Pricing discrete lookback options under a jump diffusion model

所謂回顧選擇權乃是一項契約明定投資者的報酬受到標的資產極值的影響，此種選擇權能保證給予投資者最大的報酬。大多數模型在評價回顧選擇權時，都假設連續時間觀測標的資產價格之極值，並且都能推導出精確的封閉解。然而在實務上，許多有回顧型性質的契約，都會明定特定的(離散)觀測時點，此類的選擇權即稱為離散型回顧選擇權。在本篇研究中，我們理論推倒出了在固定跳躍擴散模型(constant jump diffusion model)之下，利用修正過後之連續型回顧選擇權公式來逼近離散型回顧選擇權的價格。使用的方法參考Broadie et al. (1997)，這篇論文曾經指出，在幾何布朗運動架構之下，離散型回顧選擇權價格可以用修正後之連續型回顧選擇權公式來逼近。當中我們應用了Siegmund (1985)中的技巧來得到我們的結果。由論文最後數值分析之結果可以看出此逼近公式之高度正確性。A lookback option introduced in 1979 by Goldman et al. is a path dependent option settles based upon the maximum or minimum of the underlying price process achieved during the entire life of the option. Most models for pricing lookback options assume continuous monitoring of the extreme and have closed solutions. However, in practice, many real contracts with lookback provisions specify discrete monitoring times. Such options are called discrete lookback options. In this article, we focus on pricing discrete lookback options using continuous lookback formulas by applying a simple continuity correction under the constant jump diffusion model. We use the same method of correction as Broadie et al. (1999) which have solved such problems under the geometric Brownian motion setting. The correction is justified theoretically by applying the techniques from sequential analysis, particularly Siegmund (1985). And we also give numerical results.誌謝 ...i 中文摘要 .ii 英文摘要 ..iii 1 Introduction 1 2 Pricing Lookback Options 8 2.1 Continuous-monitoring case . . . . . . . . . ..8 2.2 Discrete-monitoring case . . . . . . . . . . .12 2.3 Continuity correction . . . . . . . . . . . . 14 3 Proofs of the Main Results 19 3.1 Distribution of the _rst passage times . . . . 19 3.2 Some known results . . . . . . . . . . . . . 21 3.3 Proof of Theorem 2.1 for the LBP case . . . .. 23 4 Numerical Results 24 4.1 Laplace transforms methods for continuous case . .24 4.2 Pricing discrete LBP and numerical results . . 28 5 Conclusion Remarks and Further Research 32 Reference 3